定积分近似计算

一、实验目的

通过本实验加深理解定积分理论中分割，近似，求和，取极限的思想方法，学习并掌握定积分的数值计算方法。

二、实验使用的软件

Mathematica 5.0或以上版本.

三、实验的基本理论及方法

1.定积分与定积分的近似计算：先对一个函数,例如sinx在区间[0,1],在程序中改变m并适当扩展有效数字,运行程序计算定积分

四、实验的内容与步骤

1.编写Mathematica程序

a=0;b=1;k=10;

f[x_]:=Sin[x];

d=N[Integrate[f[x],{x,a,b}],k];

s1[m_]:=N[Sum[f[a+i*(b-a)/m]*(b-a)/m,{i,0,m-1}],k];

s2[m_]:=N[Sum[f[a+(i+1/2)*(b-a)/m]*(b-a)/m,{i,0,m-1}],k]; s3[m_]:=N[Sum[f[a+i*(b-a)/m]*(b-a)/m,{i,1,m}],k];

s4[m_]:=N[Sum[(f[a+i*(b-a)/m]+f[a+(i+1)*(b-a)/m])/2*(b-a)/m,{i,0,m-1}],k];

s5[m_]:=N[(b-a)/m/6*((f[a]+f[b])+2*Sum[f[a+i*(b-a)/m],{i,1,m-1}]+4*Sum[f[a+(i-1/2)*(b-a)/m],{i,1,m}]),k]; r1[m_]:=d-s1[m];

r2[m_]:=d-s2[m];

r3[m_]:=d-s3[m];

r4[m_]:=d-s4[m];

r5[m_]:=d-s5[m];

t=Table[{s1[m],r1[m],s2[m],r2[m],s3[m],r3[m],s4[m],r4[m],s5[m],r5

[m]},{m,100,1000,100}]

输出结果：

{{0.455487,0.00421119,0.4597,-1.91541?0,0.463901,-0.00420352,0.459694,3.83082?0,0.459698,-1.59622?0},{0.457593,0.00210464,0.459698,-4.78852?0,0.4618,-0.00210272,0.459697,9.57704?0,0.459698,-

9.9698?0},{0.458295,0.00140288,0.459698,-2.12823?0,0.4611,-0.00-14-7-7-7-6-12-6

140203,0.459697,4.25646?0,0.459698,-1.9873?0},{0.458646,0.00105208,0.459698,-1.19713?0,0.460749,-0.0010516,0.459697,2.39426?0,0.459698,-6.66134?0},{0.458856,0.000841624,0.459698,-7.66163?0,0.460539,-0.000841318,0.459698,1.53233?0,0.459698,-2.66454?0},{0.458996,0.000701332,0.459698,-5.32058?0,0.460399,-0.000701119,0.459698,1.06412?0,0.459698,-1.27676?0},{0.459097,0.000601129,0.459698,-3.90899?0,0.460299,-0.000600973,0.459698,7.81799?0,0.459698,-8.88178?0},{0.459172,0.000525979,0.459698,-2.99282?0,0.460224,-0.00052586,0.459698,5.98565?0,0.459698,-1.11022?0},{0.45923,0.000467531,0.459698,-2.3647?0,0.460165,-0.000467437,0.459698,4.7294?0,0.459698,-4.44089?0},{0.459277,0.000420774,0.459698,-

1.91541?0,0.460118,-0.000420697,0.459698,3.83081?0,0.459698,1.66533?0}}

实验观察：当M值越来越大时，数值越来越趋近于积分值。

思考：考虑其它函数,例如y=1,y=x,y=x2,y=ex,y=ln(1+x) -16-8-8-8-16-8-8-16-16-8-8-8-7-15-8-7-15-15-8-7-7-7-14

五、实验总结

当对定积分进行计算，即求所围成的曲边梯形的面积。对其进行分割，近似，求和，取极限。当分割越细时，所求得的值就越为精确。当m值充分大时，就得到积分的近似值。

 

第二篇：matlab定积分的近似计算 实验报告二

《matlab与数学实验》实验报告

实验序号：     实验二            日期： 2015   年 05 月 09 日